src/HOL/Lifting.thy
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refactoring; generate rep_eq always, not only when it would be accepted by the code generator
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(*  Title:      HOL/Lifting.thy
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    Author:     Brian Huffman and Ondrej Kuncar
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    Author:     Cezary Kaliszyk and Christian Urban
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*)
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header {* Lifting package *}
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theory Lifting
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imports Equiv_Relations Transfer
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keywords
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  "parametric" and
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  "print_quot_maps" "print_quotients" :: diag and
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  "lift_definition" :: thy_goal and
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  "setup_lifting" "lifting_forget" "lifting_update" :: thy_decl
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begin
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subsection {* Function map *}
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context
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begin
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interpretation lifting_syntax .
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lemma map_fun_id:
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  "(id ---> id) = id"
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  by (simp add: fun_eq_iff)
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subsection {* Other predicates on relations *}
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definition left_total :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> bool"
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  where "left_total R \<longleftrightarrow> (\<forall>x. \<exists>y. R x y)"
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lemma left_totalI:
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  "(\<And>x. \<exists>y. R x y) \<Longrightarrow> left_total R"
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unfolding left_total_def by blast
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lemma left_totalE:
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  assumes "left_total R"
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  obtains "(\<And>x. \<exists>y. R x y)"
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using assms unfolding left_total_def by blast
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lemma bi_total_iff: "bi_total A = (right_total A \<and> left_total A)"
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unfolding left_total_def right_total_def bi_total_def by blast
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lemma bi_total_conv_left_right: "bi_total R \<longleftrightarrow> left_total R \<and> right_total R"
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by(simp add: left_total_def right_total_def bi_total_def)
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definition left_unique :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> bool"
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  where "left_unique R \<longleftrightarrow> (\<forall>x y z. R x z \<longrightarrow> R y z \<longrightarrow> x = y)"
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lemma left_unique_transfer [transfer_rule]:
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  assumes [transfer_rule]: "right_total A"
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  assumes [transfer_rule]: "right_total B"
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  assumes [transfer_rule]: "bi_unique A"
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  shows "((A ===> B ===> op=) ===> implies) left_unique left_unique"
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using assms unfolding left_unique_def[abs_def] right_total_def bi_unique_def fun_rel_def
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by metis
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lemma bi_unique_iff: "bi_unique A = (right_unique A \<and> left_unique A)"
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lemma bi_unique_conv_left_right: "bi_unique R \<longleftrightarrow> left_unique R \<and> right_unique R"
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by(auto simp add: left_unique_def right_unique_def bi_unique_def)
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lemma left_uniqueI: "(\<And>x y z. \<lbrakk> A x z; A y z \<rbrakk> \<Longrightarrow> x = y) \<Longrightarrow> left_unique A"
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lemma left_uniqueD: "\<lbrakk> left_unique A; A x z; A y z \<rbrakk> \<Longrightarrow> x = y"
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lemma left_total_fun:
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  "\<lbrakk>left_unique A; left_total B\<rbrakk> \<Longrightarrow> left_total (A ===> B)"
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  unfolding left_total_def fun_rel_def
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  apply (rule allI, rename_tac f)
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  apply (rule_tac x="\<lambda>y. SOME z. B (f (THE x. A x y)) z" in exI)
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  apply clarify
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  apply (subgoal_tac "(THE x. A x y) = x", simp)
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  apply (rule someI_ex)
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  apply (simp)
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  apply (rule the_equality)
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  apply assumption
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  apply (simp add: left_unique_def)
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  done
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lemma left_unique_fun:
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  "\<lbrakk>left_total A; left_unique B\<rbrakk> \<Longrightarrow> left_unique (A ===> B)"
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  unfolding left_total_def left_unique_def fun_rel_def
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  by (clarify, rule ext, fast)
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lemma left_total_eq: "left_total op=" unfolding left_total_def by blast
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lemma left_unique_eq: "left_unique op=" unfolding left_unique_def by blast
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lemma [simp]:
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  shows left_unique_conversep: "left_unique A\<inverse>\<inverse> \<longleftrightarrow> right_unique A"
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  and right_unique_conversep: "right_unique A\<inverse>\<inverse> \<longleftrightarrow> left_unique A"
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by(auto simp add: left_unique_def right_unique_def)
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lemma [simp]:
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  shows left_total_conversep: "left_total A\<inverse>\<inverse> \<longleftrightarrow> right_total A"
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  and right_total_conversep: "right_total A\<inverse>\<inverse> \<longleftrightarrow> left_total A"
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by(simp_all add: left_total_def right_total_def)
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subsection {* Quotient Predicate *}
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definition
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  "Quotient R Abs Rep T \<longleftrightarrow>
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     (\<forall>a. Abs (Rep a) = a) \<and> 
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     (\<forall>a. R (Rep a) (Rep a)) \<and>
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     (\<forall>r s. R r s \<longleftrightarrow> R r r \<and> R s s \<and> Abs r = Abs s) \<and>
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     T = (\<lambda>x y. R x x \<and> Abs x = y)"
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lemma QuotientI:
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  assumes "\<And>a. Abs (Rep a) = a"
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    and "\<And>a. R (Rep a) (Rep a)"
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    and "\<And>r s. R r s \<longleftrightarrow> R r r \<and> R s s \<and> Abs r = Abs s"
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    and "T = (\<lambda>x y. R x x \<and> Abs x = y)"
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  shows "Quotient R Abs Rep T"
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  using assms unfolding Quotient_def by blast
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context
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  fixes R Abs Rep T
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  assumes a: "Quotient R Abs Rep T"
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begin
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lemma Quotient_abs_rep: "Abs (Rep a) = a"
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  using a unfolding Quotient_def
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  by simp
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lemma Quotient_rep_reflp: "R (Rep a) (Rep a)"
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  using a unfolding Quotient_def
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  by blast
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lemma Quotient_rel:
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  "R r r \<and> R s s \<and> Abs r = Abs s \<longleftrightarrow> R r s" -- {* orientation does not loop on rewriting *}
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  using a unfolding Quotient_def
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  by blast
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lemma Quotient_cr_rel: "T = (\<lambda>x y. R x x \<and> Abs x = y)"
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  using a unfolding Quotient_def
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  by blast
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lemma Quotient_refl1: "R r s \<Longrightarrow> R r r"
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  using a unfolding Quotient_def
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  by fast
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lemma Quotient_refl2: "R r s \<Longrightarrow> R s s"
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  using a unfolding Quotient_def
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  by fast
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lemma Quotient_rel_rep: "R (Rep a) (Rep b) \<longleftrightarrow> a = b"
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  using a unfolding Quotient_def
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  by metis
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lemma Quotient_rep_abs: "R r r \<Longrightarrow> R (Rep (Abs r)) r"
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  using a unfolding Quotient_def
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  by blast
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lemma Quotient_rep_abs_eq: "R t t \<Longrightarrow> R \<le> op= \<Longrightarrow> Rep (Abs t) = t"
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  using a unfolding Quotient_def
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  by blast
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lemma Quotient_rep_abs_fold_unmap: 
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  assumes "x' \<equiv> Abs x" and "R x x" and "Rep x' \<equiv> Rep' x'" 
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  shows "R (Rep' x') x"
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proof -
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  have "R (Rep x') x" using assms(1-2) Quotient_rep_abs by auto
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  then show ?thesis using assms(3) by simp
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qed
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lemma Quotient_Rep_eq:
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  assumes "x' \<equiv> Abs x" 
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  shows "Rep x' \<equiv> Rep x'"
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by simp
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lemma Quotient_rel_abs: "R r s \<Longrightarrow> Abs r = Abs s"
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  using a unfolding Quotient_def
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  by blast
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lemma Quotient_rel_abs2:
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  assumes "R (Rep x) y"
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  shows "x = Abs y"
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proof -
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  from assms have "Abs (Rep x) = Abs y" by (auto intro: Quotient_rel_abs)
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  then show ?thesis using assms(1) by (simp add: Quotient_abs_rep)
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qed
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lemma Quotient_symp: "symp R"
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  using a unfolding Quotient_def using sympI by (metis (full_types))
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lemma Quotient_transp: "transp R"
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  using a unfolding Quotient_def using transpI by (metis (full_types))
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lemma Quotient_part_equivp: "part_equivp R"
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by (metis Quotient_rep_reflp Quotient_symp Quotient_transp part_equivpI)
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end
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lemma identity_quotient: "Quotient (op =) id id (op =)"
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unfolding Quotient_def by simp 
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text {* TODO: Use one of these alternatives as the real definition. *}
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lemma Quotient_alt_def:
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  "Quotient R Abs Rep T \<longleftrightarrow>
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    (\<forall>a b. T a b \<longrightarrow> Abs a = b) \<and>
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    (\<forall>b. T (Rep b) b) \<and>
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    (\<forall>x y. R x y \<longleftrightarrow> T x (Abs x) \<and> T y (Abs y) \<and> Abs x = Abs y)"
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apply safe
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apply (simp (no_asm_use) only: Quotient_def, fast)
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apply (simp (no_asm_use) only: Quotient_def, fast)
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apply (simp (no_asm_use) only: Quotient_def, fast)
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apply (simp (no_asm_use) only: Quotient_def, fast)
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apply (simp (no_asm_use) only: Quotient_def, fast)
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apply (simp (no_asm_use) only: Quotient_def, fast)
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apply (rule QuotientI)
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apply simp
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apply metis
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apply simp
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apply (rule ext, rule ext, metis)
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done
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lemma Quotient_alt_def2:
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  "Quotient R Abs Rep T \<longleftrightarrow>
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    (\<forall>a b. T a b \<longrightarrow> Abs a = b) \<and>
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    (\<forall>b. T (Rep b) b) \<and>
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    (\<forall>x y. R x y \<longleftrightarrow> T x (Abs y) \<and> T y (Abs x))"
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  unfolding Quotient_alt_def by (safe, metis+)
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lemma Quotient_alt_def3:
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  "Quotient R Abs Rep T \<longleftrightarrow>
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    (\<forall>a b. T a b \<longrightarrow> Abs a = b) \<and> (\<forall>b. T (Rep b) b) \<and>
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    (\<forall>x y. R x y \<longleftrightarrow> (\<exists>z. T x z \<and> T y z))"
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  unfolding Quotient_alt_def2 by (safe, metis+)
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lemma Quotient_alt_def4:
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  "Quotient R Abs Rep T \<longleftrightarrow>
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    (\<forall>a b. T a b \<longrightarrow> Abs a = b) \<and> (\<forall>b. T (Rep b) b) \<and> R = T OO conversep T"
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  unfolding Quotient_alt_def3 fun_eq_iff by auto
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lemma fun_quotient:
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  assumes 1: "Quotient R1 abs1 rep1 T1"
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  assumes 2: "Quotient R2 abs2 rep2 T2"
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  shows "Quotient (R1 ===> R2) (rep1 ---> abs2) (abs1 ---> rep2) (T1 ===> T2)"
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  using assms unfolding Quotient_alt_def2
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  unfolding fun_rel_def fun_eq_iff map_fun_apply
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  by (safe, metis+)
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lemma apply_rsp:
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  fixes f g::"'a \<Rightarrow> 'c"
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  assumes q: "Quotient R1 Abs1 Rep1 T1"
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  and     a: "(R1 ===> R2) f g" "R1 x y"
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  shows "R2 (f x) (g y)"
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  using a by (auto elim: fun_relE)
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lemma apply_rsp':
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  assumes a: "(R1 ===> R2) f g" "R1 x y"
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  shows "R2 (f x) (g y)"
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  using a by (auto elim: fun_relE)
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lemma apply_rsp'':
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  assumes "Quotient R Abs Rep T"
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  and "(R ===> S) f f"
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  shows "S (f (Rep x)) (f (Rep x))"
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proof -
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  from assms(1) have "R (Rep x) (Rep x)" by (rule Quotient_rep_reflp)
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  then show ?thesis using assms(2) by (auto intro: apply_rsp')
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qed
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subsection {* Quotient composition *}
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lemma Quotient_compose:
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  assumes 1: "Quotient R1 Abs1 Rep1 T1"
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  assumes 2: "Quotient R2 Abs2 Rep2 T2"
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  shows "Quotient (T1 OO R2 OO conversep T1) (Abs2 \<circ> Abs1) (Rep1 \<circ> Rep2) (T1 OO T2)"
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  using assms unfolding Quotient_alt_def4 by fastforce
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lemma equivp_reflp2:
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  "equivp R \<Longrightarrow> reflp R"
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  by (erule equivpE)
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subsection {* Respects predicate *}
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definition Respects :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a set"
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  where "Respects R = {x. R x x}"
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e455cdaac479 move constant 'Respects' into Lifting.thy;
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lemma in_respects: "x \<in> Respects R \<longleftrightarrow> R x x"
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  unfolding Respects_def by simp
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subsection {* Invariant *}
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definition invariant :: "('a \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool" 
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  where "invariant R = (\<lambda>x y. R x \<and> x = y)"
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lemma invariant_to_eq:
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  assumes "invariant P x y"
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  shows "x = y"
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using assms by (simp add: invariant_def)
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lemma fun_rel_eq_invariant:
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  shows "((invariant R) ===> S) = (\<lambda>f g. \<forall>x. R x \<longrightarrow> S (f x) (g x))"
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by (auto simp add: invariant_def fun_rel_def)
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lemma invariant_same_args:
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  shows "invariant P x x \<equiv> P x"
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using assms by (auto simp add: invariant_def)
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lemma invariant_transfer [transfer_rule]:
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  assumes [transfer_rule]: "bi_unique A"
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  shows "((A ===> op=) ===> A ===> A ===> op=) Lifting.invariant Lifting.invariant"
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unfolding invariant_def[abs_def] by transfer_prover
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lemma UNIV_typedef_to_Quotient:
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  assumes "type_definition Rep Abs UNIV"
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  and T_def: "T \<equiv> (\<lambda>x y. x = Rep y)"
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parents:
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   315
  shows "Quotient (op =) Abs Rep T"
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parents:
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   316
proof -
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parents:
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   317
  interpret type_definition Rep Abs UNIV by fact
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  from Abs_inject Rep_inverse Abs_inverse T_def show ?thesis 
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   319
    by (fastforce intro!: QuotientI fun_eq_iff)
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parents:
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   320
qed
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parents:
diff changeset
   321
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   322
lemma UNIV_typedef_to_equivp:
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parents:
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   323
  fixes Abs :: "'a \<Rightarrow> 'b"
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parents:
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   324
  and Rep :: "'b \<Rightarrow> 'a"
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parents:
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   325
  assumes "type_definition Rep Abs (UNIV::'a set)"
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kuncar
parents:
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   326
  shows "equivp (op=::'a\<Rightarrow>'a\<Rightarrow>bool)"
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parents:
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   327
by (rule identity_equivp)
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parents:
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   328
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   329
lemma typedef_to_Quotient:
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   330
  assumes "type_definition Rep Abs S"
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   331
  and T_def: "T \<equiv> (\<lambda>x y. x = Rep y)"
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   332
  shows "Quotient (invariant (\<lambda>x. x \<in> S)) Abs Rep T"
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kuncar
parents: 47354
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   333
proof -
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kuncar
parents: 47354
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   334
  interpret type_definition Rep Abs S by fact
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   335
  from Rep Abs_inject Rep_inverse Abs_inverse T_def show ?thesis
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   336
    by (auto intro!: QuotientI simp: invariant_def fun_eq_iff)
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   337
qed
87c0eaf04bad support non-open typedefs; define cr_rel in terms of a rep function for typedefs
kuncar
parents: 47354
diff changeset
   338
87c0eaf04bad support non-open typedefs; define cr_rel in terms of a rep function for typedefs
kuncar
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   339
lemma typedef_to_part_equivp:
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   340
  assumes "type_definition Rep Abs S"
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parents: 47436
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   341
  shows "part_equivp (invariant (\<lambda>x. x \<in> S))"
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kuncar
parents: 47354
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   342
proof (intro part_equivpI)
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   343
  interpret type_definition Rep Abs S by fact
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   344
  show "\<exists>x. invariant (\<lambda>x. x \<in> S) x x" using Rep by (auto simp: invariant_def)
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   345
next
47501
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parents: 47436
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   346
  show "symp (invariant (\<lambda>x. x \<in> S))" by (auto intro: sympI simp: invariant_def)
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   347
next
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parents: 47436
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   348
  show "transp (invariant (\<lambda>x. x \<in> S))" by (auto intro: transpI simp: invariant_def)
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kuncar
parents: 47354
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   349
qed
87c0eaf04bad support non-open typedefs; define cr_rel in terms of a rep function for typedefs
kuncar
parents: 47354
diff changeset
   350
87c0eaf04bad support non-open typedefs; define cr_rel in terms of a rep function for typedefs
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   351
lemma open_typedef_to_Quotient:
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parents:
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   352
  assumes "type_definition Rep Abs {x. P x}"
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parents: 47351
diff changeset
   353
  and T_def: "T \<equiv> (\<lambda>x y. x = Rep y)"
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parents:
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   354
  shows "Quotient (invariant P) Abs Rep T"
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parents: 47575
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   355
  using typedef_to_Quotient [OF assms] by simp
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parents:
diff changeset
   356
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   357
lemma open_typedef_to_part_equivp:
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parents:
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   358
  assumes "type_definition Rep Abs {x. P x}"
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   359
  shows "part_equivp (invariant P)"
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huffman
parents: 47575
diff changeset
   360
  using typedef_to_part_equivp [OF assms] by simp
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parents:
diff changeset
   361
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   362
text {* Generating transfer rules for quotients. *}
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diff changeset
   363
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diff changeset
   364
context
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parents: 47536
diff changeset
   365
  fixes R Abs Rep T
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   366
  assumes 1: "Quotient R Abs Rep T"
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   367
begin
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   368
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   369
lemma Quotient_right_unique: "right_unique T"
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   370
  using 1 unfolding Quotient_alt_def right_unique_def by metis
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parents: 47536
diff changeset
   371
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   372
lemma Quotient_right_total: "right_total T"
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   373
  using 1 unfolding Quotient_alt_def right_total_def by metis
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parents: 47536
diff changeset
   374
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   375
lemma Quotient_rel_eq_transfer: "(T ===> T ===> op =) R (op =)"
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parents: 47536
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   376
  using 1 unfolding Quotient_alt_def fun_rel_def by simp
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parents: 47369
diff changeset
   377
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   378
lemma Quotient_abs_induct:
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   379
  assumes "\<And>y. R y y \<Longrightarrow> P (Abs y)" shows "P x"
1f0ec5b8135a add lemma Quotient_abs_induct
huffman
parents: 47537
diff changeset
   380
  using 1 assms unfolding Quotient_def by metis
1f0ec5b8135a add lemma Quotient_abs_induct
huffman
parents: 47537
diff changeset
   381
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diff changeset
   382
end
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diff changeset
   383
b06be48923a4 more usage of context blocks
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   384
text {* Generating transfer rules for total quotients. *}
47376
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huffman
parents: 47369
diff changeset
   385
47537
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huffman
parents: 47536
diff changeset
   386
context
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huffman
parents: 47536
diff changeset
   387
  fixes R Abs Rep T
b06be48923a4 more usage of context blocks
huffman
parents: 47536
diff changeset
   388
  assumes 1: "Quotient R Abs Rep T" and 2: "reflp R"
b06be48923a4 more usage of context blocks
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   389
begin
47376
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huffman
parents: 47369
diff changeset
   390
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   391
lemma Quotient_bi_total: "bi_total T"
b06be48923a4 more usage of context blocks
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parents: 47536
diff changeset
   392
  using 1 2 unfolding Quotient_alt_def bi_total_def reflp_def by auto
b06be48923a4 more usage of context blocks
huffman
parents: 47536
diff changeset
   393
b06be48923a4 more usage of context blocks
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diff changeset
   394
lemma Quotient_id_abs_transfer: "(op = ===> T) (\<lambda>x. x) Abs"
b06be48923a4 more usage of context blocks
huffman
parents: 47536
diff changeset
   395
  using 1 2 unfolding Quotient_alt_def reflp_def fun_rel_def by simp
b06be48923a4 more usage of context blocks
huffman
parents: 47536
diff changeset
   396
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b90cd7016d4f generate abs_induct rules for quotient types
huffman
parents: 47545
diff changeset
   397
lemma Quotient_total_abs_induct: "(\<And>y. P (Abs y)) \<Longrightarrow> P x"
b90cd7016d4f generate abs_induct rules for quotient types
huffman
parents: 47545
diff changeset
   398
  using 1 2 assms unfolding Quotient_alt_def reflp_def by metis
b90cd7016d4f generate abs_induct rules for quotient types
huffman
parents: 47545
diff changeset
   399
47889
29212a4bb866 lifting package produces abs_eq_iff rules for total quotients
huffman
parents: 47777
diff changeset
   400
lemma Quotient_total_abs_eq_iff: "Abs x = Abs y \<longleftrightarrow> R x y"
29212a4bb866 lifting package produces abs_eq_iff rules for total quotients
huffman
parents: 47777
diff changeset
   401
  using Quotient_rel [OF 1] 2 unfolding reflp_def by simp
29212a4bb866 lifting package produces abs_eq_iff rules for total quotients
huffman
parents: 47777
diff changeset
   402
47537
b06be48923a4 more usage of context blocks
huffman
parents: 47536
diff changeset
   403
end
47376
776254f89a18 add transfer lemmas for quotients
huffman
parents: 47369
diff changeset
   404
47368
4c522dff1f4d add lemmas for generating transfer rules for typedefs
huffman
parents: 47354
diff changeset
   405
text {* Generating transfer rules for a type defined with @{text "typedef"}. *}
4c522dff1f4d add lemmas for generating transfer rules for typedefs
huffman
parents: 47354
diff changeset
   406
47534
06cc372a80ed use context block to organize typedef lifting theorems
huffman
parents: 47521
diff changeset
   407
context
06cc372a80ed use context block to organize typedef lifting theorems
huffman
parents: 47521
diff changeset
   408
  fixes Rep Abs A T
47368
4c522dff1f4d add lemmas for generating transfer rules for typedefs
huffman
parents: 47354
diff changeset
   409
  assumes type: "type_definition Rep Abs A"
47534
06cc372a80ed use context block to organize typedef lifting theorems
huffman
parents: 47521
diff changeset
   410
  assumes T_def: "T \<equiv> (\<lambda>(x::'a) (y::'b). x = Rep y)"
06cc372a80ed use context block to organize typedef lifting theorems
huffman
parents: 47521
diff changeset
   411
begin
06cc372a80ed use context block to organize typedef lifting theorems
huffman
parents: 47521
diff changeset
   412
51994
82cc2aeb7d13 stronger reflexivity prover
kuncar
parents: 51956
diff changeset
   413
lemma typedef_left_unique: "left_unique T"
82cc2aeb7d13 stronger reflexivity prover
kuncar
parents: 51956
diff changeset
   414
  unfolding left_unique_def T_def
82cc2aeb7d13 stronger reflexivity prover
kuncar
parents: 51956
diff changeset
   415
  by (simp add: type_definition.Rep_inject [OF type])
82cc2aeb7d13 stronger reflexivity prover
kuncar
parents: 51956
diff changeset
   416
47534
06cc372a80ed use context block to organize typedef lifting theorems
huffman
parents: 47521
diff changeset
   417
lemma typedef_bi_unique: "bi_unique T"
47368
4c522dff1f4d add lemmas for generating transfer rules for typedefs
huffman
parents: 47354
diff changeset
   418
  unfolding bi_unique_def T_def
4c522dff1f4d add lemmas for generating transfer rules for typedefs
huffman
parents: 47354
diff changeset
   419
  by (simp add: type_definition.Rep_inject [OF type])
4c522dff1f4d add lemmas for generating transfer rules for typedefs
huffman
parents: 47354
diff changeset
   420
51374
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   421
(* the following two theorems are here only for convinience *)
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   422
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   423
lemma typedef_right_unique: "right_unique T"
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   424
  using T_def type Quotient_right_unique typedef_to_Quotient 
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   425
  by blast
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   426
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   427
lemma typedef_right_total: "right_total T"
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   428
  using T_def type Quotient_right_total typedef_to_Quotient 
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   429
  by blast
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   430
47535
0f94b02fda1c lifting_setup generates transfer rule for rep of typedefs
huffman
parents: 47534
diff changeset
   431
lemma typedef_rep_transfer: "(T ===> op =) (\<lambda>x. x) Rep"
0f94b02fda1c lifting_setup generates transfer rule for rep of typedefs
huffman
parents: 47534
diff changeset
   432
  unfolding fun_rel_def T_def by simp
0f94b02fda1c lifting_setup generates transfer rule for rep of typedefs
huffman
parents: 47534
diff changeset
   433
47534
06cc372a80ed use context block to organize typedef lifting theorems
huffman
parents: 47521
diff changeset
   434
end
06cc372a80ed use context block to organize typedef lifting theorems
huffman
parents: 47521
diff changeset
   435
47368
4c522dff1f4d add lemmas for generating transfer rules for typedefs
huffman
parents: 47354
diff changeset
   436
text {* Generating the correspondence rule for a constant defined with
4c522dff1f4d add lemmas for generating transfer rules for typedefs
huffman
parents: 47354
diff changeset
   437
  @{text "lift_definition"}. *}
4c522dff1f4d add lemmas for generating transfer rules for typedefs
huffman
parents: 47354
diff changeset
   438
47351
0193e663a19e lift_definition command generates transfer rule
huffman
parents: 47325
diff changeset
   439
lemma Quotient_to_transfer:
0193e663a19e lift_definition command generates transfer rule
huffman
parents: 47325
diff changeset
   440
  assumes "Quotient R Abs Rep T" and "R c c" and "c' \<equiv> Abs c"
0193e663a19e lift_definition command generates transfer rule
huffman
parents: 47325
diff changeset
   441
  shows "T c c'"
0193e663a19e lift_definition command generates transfer rule
huffman
parents: 47325
diff changeset
   442
  using assms by (auto dest: Quotient_cr_rel)
0193e663a19e lift_definition command generates transfer rule
huffman
parents: 47325
diff changeset
   443
47982
7aa35601ff65 prove reflexivity also for the quotient composition relation; reflp_preserve renamed to reflexivity_rule
kuncar
parents: 47937
diff changeset
   444
text {* Proving reflexivity *}
7aa35601ff65 prove reflexivity also for the quotient composition relation; reflp_preserve renamed to reflexivity_rule
kuncar
parents: 47937
diff changeset
   445
7aa35601ff65 prove reflexivity also for the quotient composition relation; reflp_preserve renamed to reflexivity_rule
kuncar
parents: 47937
diff changeset
   446
lemma Quotient_to_left_total:
7aa35601ff65 prove reflexivity also for the quotient composition relation; reflp_preserve renamed to reflexivity_rule
kuncar
parents: 47937
diff changeset
   447
  assumes q: "Quotient R Abs Rep T"
7aa35601ff65 prove reflexivity also for the quotient composition relation; reflp_preserve renamed to reflexivity_rule
kuncar
parents: 47937
diff changeset
   448
  and r_R: "reflp R"
7aa35601ff65 prove reflexivity also for the quotient composition relation; reflp_preserve renamed to reflexivity_rule
kuncar
parents: 47937
diff changeset
   449
  shows "left_total T"
7aa35601ff65 prove reflexivity also for the quotient composition relation; reflp_preserve renamed to reflexivity_rule
kuncar
parents: 47937
diff changeset
   450
using r_R Quotient_cr_rel[OF q] unfolding left_total_def by (auto elim: reflpE)
7aa35601ff65 prove reflexivity also for the quotient composition relation; reflp_preserve renamed to reflexivity_rule
kuncar
parents: 47937
diff changeset
   451
55563
a64d49f49ca3 implement the reflexivity prover as a monotonicity prover that proves R >= op=; derive "reflexivity" rules for relators from mono rules and eq rules
kuncar
parents: 55083
diff changeset
   452
lemma Quotient_composition_ge_eq:
a64d49f49ca3 implement the reflexivity prover as a monotonicity prover that proves R >= op=; derive "reflexivity" rules for relators from mono rules and eq rules
kuncar
parents: 55083
diff changeset
   453
  assumes "left_total T"
a64d49f49ca3 implement the reflexivity prover as a monotonicity prover that proves R >= op=; derive "reflexivity" rules for relators from mono rules and eq rules
kuncar
parents: 55083
diff changeset
   454
  assumes "R \<ge> op="
a64d49f49ca3 implement the reflexivity prover as a monotonicity prover that proves R >= op=; derive "reflexivity" rules for relators from mono rules and eq rules
kuncar
parents: 55083
diff changeset
   455
  shows "(T OO R OO T\<inverse>\<inverse>) \<ge> op="
a64d49f49ca3 implement the reflexivity prover as a monotonicity prover that proves R >= op=; derive "reflexivity" rules for relators from mono rules and eq rules
kuncar
parents: 55083
diff changeset
   456
using assms unfolding left_total_def by fast
51994
82cc2aeb7d13 stronger reflexivity prover
kuncar
parents: 51956
diff changeset
   457
55563
a64d49f49ca3 implement the reflexivity prover as a monotonicity prover that proves R >= op=; derive "reflexivity" rules for relators from mono rules and eq rules
kuncar
parents: 55083
diff changeset
   458
lemma Quotient_composition_le_eq:
a64d49f49ca3 implement the reflexivity prover as a monotonicity prover that proves R >= op=; derive "reflexivity" rules for relators from mono rules and eq rules
kuncar
parents: 55083
diff changeset
   459
  assumes "left_unique T"
a64d49f49ca3 implement the reflexivity prover as a monotonicity prover that proves R >= op=; derive "reflexivity" rules for relators from mono rules and eq rules
kuncar
parents: 55083
diff changeset
   460
  assumes "R \<le> op="
a64d49f49ca3 implement the reflexivity prover as a monotonicity prover that proves R >= op=; derive "reflexivity" rules for relators from mono rules and eq rules
kuncar
parents: 55083
diff changeset
   461
  shows "(T OO R OO T\<inverse>\<inverse>) \<le> op="
55604
42e4e8c2e8dc less flex-flex pairs
noschinl
parents: 55563
diff changeset
   462
using assms unfolding left_unique_def by blast
47982
7aa35601ff65 prove reflexivity also for the quotient composition relation; reflp_preserve renamed to reflexivity_rule
kuncar
parents: 47937
diff changeset
   463
52307
32c433c38ddd more reflexivity rules (for OO)
kuncar
parents: 52036
diff changeset
   464
lemma left_total_composition: "left_total R \<Longrightarrow> left_total S \<Longrightarrow> left_total (R OO S)"
32c433c38ddd more reflexivity rules (for OO)
kuncar
parents: 52036
diff changeset
   465
unfolding left_total_def OO_def by fast
32c433c38ddd more reflexivity rules (for OO)
kuncar
parents: 52036
diff changeset
   466
32c433c38ddd more reflexivity rules (for OO)
kuncar
parents: 52036
diff changeset
   467
lemma left_unique_composition: "left_unique R \<Longrightarrow> left_unique S \<Longrightarrow> left_unique (R OO S)"
55604
42e4e8c2e8dc less flex-flex pairs
noschinl
parents: 55563
diff changeset
   468
unfolding left_unique_def OO_def by blast
52307
32c433c38ddd more reflexivity rules (for OO)
kuncar
parents: 52036
diff changeset
   469
55563
a64d49f49ca3 implement the reflexivity prover as a monotonicity prover that proves R >= op=; derive "reflexivity" rules for relators from mono rules and eq rules
kuncar
parents: 55083
diff changeset
   470
lemma invariant_le_eq:
a64d49f49ca3 implement the reflexivity prover as a monotonicity prover that proves R >= op=; derive "reflexivity" rules for relators from mono rules and eq rules
kuncar
parents: 55083
diff changeset
   471
  "invariant P \<le> op=" unfolding invariant_def by blast
a64d49f49ca3 implement the reflexivity prover as a monotonicity prover that proves R >= op=; derive "reflexivity" rules for relators from mono rules and eq rules
kuncar
parents: 55083
diff changeset
   472
a64d49f49ca3 implement the reflexivity prover as a monotonicity prover that proves R >= op=; derive "reflexivity" rules for relators from mono rules and eq rules
kuncar
parents: 55083
diff changeset
   473
lemma reflp_ge_eq:
a64d49f49ca3 implement the reflexivity prover as a monotonicity prover that proves R >= op=; derive "reflexivity" rules for relators from mono rules and eq rules
kuncar
parents: 55083
diff changeset
   474
  "reflp R \<Longrightarrow> R \<ge> op=" unfolding reflp_def by blast
a64d49f49ca3 implement the reflexivity prover as a monotonicity prover that proves R >= op=; derive "reflexivity" rules for relators from mono rules and eq rules
kuncar
parents: 55083
diff changeset
   475
a64d49f49ca3 implement the reflexivity prover as a monotonicity prover that proves R >= op=; derive "reflexivity" rules for relators from mono rules and eq rules
kuncar
parents: 55083
diff changeset
   476
lemma ge_eq_refl:
a64d49f49ca3 implement the reflexivity prover as a monotonicity prover that proves R >= op=; derive "reflexivity" rules for relators from mono rules and eq rules
kuncar
parents: 55083
diff changeset
   477
  "R \<ge> op= \<Longrightarrow> R x x" by blast
47982
7aa35601ff65 prove reflexivity also for the quotient composition relation; reflp_preserve renamed to reflexivity_rule
kuncar
parents: 47937
diff changeset
   478
51374
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   479
text {* Proving a parametrized correspondence relation *}
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   480
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   481
definition POS :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> bool" where
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   482
"POS A B \<equiv> A \<le> B"
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   483
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   484
definition  NEG :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> bool" where
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   485
"NEG A B \<equiv> B \<le> A"
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   486
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   487
(*
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   488
  The following two rules are here because we don't have any proper
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   489
  left-unique ant left-total relations. Left-unique and left-total
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   490
  assumptions show up in distributivity rules for the function type.
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   491
*)
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   492
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   493
lemma bi_unique_left_unique[transfer_rule]: "bi_unique R \<Longrightarrow> left_unique R"
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   494
unfolding bi_unique_def left_unique_def by blast
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   495
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   496
lemma bi_total_left_total[transfer_rule]: "bi_total R \<Longrightarrow> left_total R"
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   497
unfolding bi_total_def left_total_def by blast
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   498
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   499
lemma pos_OO_eq:
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   500
  shows "POS (A OO op=) A"
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   501
unfolding POS_def OO_def by blast
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   502
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   503
lemma pos_eq_OO:
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   504
  shows "POS (op= OO A) A"
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   505
unfolding POS_def OO_def by blast
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   506
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   507
lemma neg_OO_eq:
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   508
  shows "NEG (A OO op=) A"
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   509
unfolding NEG_def OO_def by auto
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   510
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   511
lemma neg_eq_OO:
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   512
  shows "NEG (op= OO A) A"
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   513
unfolding NEG_def OO_def by blast
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   514
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   515
lemma POS_trans:
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   516
  assumes "POS A B"
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   517
  assumes "POS B C"
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   518
  shows "POS A C"
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   519
using assms unfolding POS_def by auto
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   520
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   521
lemma NEG_trans:
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   522
  assumes "NEG A B"
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   523
  assumes "NEG B C"
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   524
  shows "NEG A C"
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   525
using assms unfolding NEG_def by auto
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   526
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   527
lemma POS_NEG:
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   528
  "POS A B \<equiv> NEG B A"
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   529
  unfolding POS_def NEG_def by auto
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   530
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   531
lemma NEG_POS:
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   532
  "NEG A B \<equiv> POS B A"
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   533
  unfolding POS_def NEG_def by auto
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   534
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   535
lemma POS_pcr_rule:
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   536
  assumes "POS (A OO B) C"
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   537
  shows "POS (A OO B OO X) (C OO X)"
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   538
using assms unfolding POS_def OO_def by blast
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   539
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   540
lemma NEG_pcr_rule:
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   541
  assumes "NEG (A OO B) C"
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   542
  shows "NEG (A OO B OO X) (C OO X)"
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   543
using assms unfolding NEG_def OO_def by blast
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   544
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   545
lemma POS_apply:
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   546
  assumes "POS R R'"
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   547
  assumes "R f g"
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   548
  shows "R' f g"
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   549
using assms unfolding POS_def by auto
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   550
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   551
text {* Proving a parametrized correspondence relation *}
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   552
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   553
lemma fun_mono:
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   554
  assumes "A \<ge> C"
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   555
  assumes "B \<le> D"
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   556
  shows   "(A ===> B) \<le> (C ===> D)"
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   557
using assms unfolding fun_rel_def by blast
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   558
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   559
lemma pos_fun_distr: "((R ===> S) OO (R' ===> S')) \<le> ((R OO R') ===> (S OO S'))"
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   560
unfolding OO_def fun_rel_def by blast
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   561
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   562
lemma functional_relation: "right_unique R \<Longrightarrow> left_total R \<Longrightarrow> \<forall>x. \<exists>!y. R x y"
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   563
unfolding right_unique_def left_total_def by blast
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   564
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   565
lemma functional_converse_relation: "left_unique R \<Longrightarrow> right_total R \<Longrightarrow> \<forall>y. \<exists>!x. R x y"
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   566
unfolding left_unique_def right_total_def by blast
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   567
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   568
lemma neg_fun_distr1:
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   569
assumes 1: "left_unique R" "right_total R"
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   570
assumes 2: "right_unique R'" "left_total R'"
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   571
shows "(R OO R' ===> S OO S') \<le> ((R ===> S) OO (R' ===> S')) "
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   572
  using functional_relation[OF 2] functional_converse_relation[OF 1]
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   573
  unfolding fun_rel_def OO_def
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   574
  apply clarify
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   575
  apply (subst all_comm)
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   576
  apply (subst all_conj_distrib[symmetric])
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   577
  apply (intro choice)
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   578
  by metis
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   579
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   580
lemma neg_fun_distr2:
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   581
assumes 1: "right_unique R'" "left_total R'"
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   582
assumes 2: "left_unique S'" "right_total S'"
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   583
shows "(R OO R' ===> S OO S') \<le> ((R ===> S) OO (R' ===> S'))"
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   584
  using functional_converse_relation[OF 2] functional_relation[OF 1]
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   585
  unfolding fun_rel_def OO_def
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   586
  apply clarify
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   587
  apply (subst all_comm)
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   588
  apply (subst all_conj_distrib[symmetric])
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   589
  apply (intro choice)
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   590
  by metis
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   591
51956
a4d81cdebf8b better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents: 51374
diff changeset
   592
subsection {* Domains *}
a4d81cdebf8b better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents: 51374
diff changeset
   593
a4d81cdebf8b better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents: 51374
diff changeset
   594
lemma pcr_Domainp_par_left_total:
a4d81cdebf8b better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents: 51374
diff changeset
   595
  assumes "Domainp B = P"
a4d81cdebf8b better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents: 51374
diff changeset
   596
  assumes "left_total A"
a4d81cdebf8b better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents: 51374
diff changeset
   597
  assumes "(A ===> op=) P' P"
a4d81cdebf8b better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents: 51374
diff changeset
   598
  shows "Domainp (A OO B) = P'"
a4d81cdebf8b better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents: 51374
diff changeset
   599
using assms
a4d81cdebf8b better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents: 51374
diff changeset
   600
unfolding Domainp_iff[abs_def] OO_def bi_unique_def left_total_def fun_rel_def 
a4d81cdebf8b better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents: 51374
diff changeset
   601
by (fast intro: fun_eq_iff)
a4d81cdebf8b better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents: 51374
diff changeset
   602
a4d81cdebf8b better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents: 51374
diff changeset
   603
lemma pcr_Domainp_par:
a4d81cdebf8b better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents: 51374
diff changeset
   604
assumes "Domainp B = P2"
a4d81cdebf8b better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents: 51374
diff changeset
   605
assumes "Domainp A = P1"
a4d81cdebf8b better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents: 51374
diff changeset
   606
assumes "(A ===> op=) P2' P2"
a4d81cdebf8b better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents: 51374
diff changeset
   607
shows "Domainp (A OO B) = (inf P1 P2')"
a4d81cdebf8b better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents: 51374
diff changeset
   608
using assms unfolding fun_rel_def Domainp_iff[abs_def] OO_def
a4d81cdebf8b better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents: 51374
diff changeset
   609
by (fast intro: fun_eq_iff)
a4d81cdebf8b better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents: 51374
diff changeset
   610
53151
fbf4d50dec91 remove OP
kuncar
parents: 53011
diff changeset
   611
definition rel_pred_comp :: "('a => 'b => bool) => ('b => bool) => 'a => bool"
51956
a4d81cdebf8b better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents: 51374
diff changeset
   612
where "rel_pred_comp R P \<equiv> \<lambda>x. \<exists>y. R x y \<and> P y"
a4d81cdebf8b better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents: 51374
diff changeset
   613
a4d81cdebf8b better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents: 51374
diff changeset
   614
lemma pcr_Domainp:
a4d81cdebf8b better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents: 51374
diff changeset
   615
assumes "Domainp B = P"
53151
fbf4d50dec91 remove OP
kuncar
parents: 53011
diff changeset
   616
shows "Domainp (A OO B) = (\<lambda>x. \<exists>y. A x y \<and> P y)"
fbf4d50dec91 remove OP
kuncar
parents: 53011
diff changeset
   617
using assms by blast
51956
a4d81cdebf8b better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents: 51374
diff changeset
   618
a4d81cdebf8b better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents: 51374
diff changeset
   619
lemma pcr_Domainp_total:
a4d81cdebf8b better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents: 51374
diff changeset
   620
  assumes "bi_total B"
a4d81cdebf8b better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents: 51374
diff changeset
   621
  assumes "Domainp A = P"
a4d81cdebf8b better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents: 51374
diff changeset
   622
  shows "Domainp (A OO B) = P"
a4d81cdebf8b better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents: 51374
diff changeset
   623
using assms unfolding bi_total_def 
a4d81cdebf8b better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents: 51374
diff changeset
   624
by fast
a4d81cdebf8b better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents: 51374
diff changeset
   625
a4d81cdebf8b better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents: 51374
diff changeset
   626
lemma Quotient_to_Domainp:
a4d81cdebf8b better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents: 51374
diff changeset
   627
  assumes "Quotient R Abs Rep T"
a4d81cdebf8b better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents: 51374
diff changeset
   628
  shows "Domainp T = (\<lambda>x. R x x)"  
a4d81cdebf8b better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents: 51374
diff changeset
   629
by (simp add: Domainp_iff[abs_def] Quotient_cr_rel[OF assms])
a4d81cdebf8b better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents: 51374
diff changeset
   630
a4d81cdebf8b better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents: 51374
diff changeset
   631
lemma invariant_to_Domainp:
a4d81cdebf8b better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents: 51374
diff changeset
   632
  assumes "Quotient (Lifting.invariant P) Abs Rep T"
a4d81cdebf8b better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents: 51374
diff changeset
   633
  shows "Domainp T = P"
a4d81cdebf8b better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents: 51374
diff changeset
   634
by (simp add: invariant_def Domainp_iff[abs_def] Quotient_cr_rel[OF assms])
a4d81cdebf8b better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents: 51374
diff changeset
   635
53011
aeee0a4be6cf introduce locale with syntax for fun_rel and map_fun and make thus ===> and ---> local
kuncar
parents: 52307
diff changeset
   636
end
aeee0a4be6cf introduce locale with syntax for fun_rel and map_fun and make thus ===> and ---> local
kuncar
parents: 52307
diff changeset
   637
47308
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   638
subsection {* ML setup *}
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   639
48891
c0eafbd55de3 prefer ML_file over old uses;
wenzelm
parents: 47982
diff changeset
   640
ML_file "Tools/Lifting/lifting_util.ML"
47308
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   641
48891
c0eafbd55de3 prefer ML_file over old uses;
wenzelm
parents: 47982
diff changeset
   642
ML_file "Tools/Lifting/lifting_info.ML"
47308
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   643
setup Lifting_Info.setup
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   644
51994
82cc2aeb7d13 stronger reflexivity prover
kuncar
parents: 51956
diff changeset
   645
lemmas [reflexivity_rule] = 
55563
a64d49f49ca3 implement the reflexivity prover as a monotonicity prover that proves R >= op=; derive "reflexivity" rules for relators from mono rules and eq rules
kuncar
parents: 55083
diff changeset
   646
  order_refl[of "op="] invariant_le_eq Quotient_composition_le_eq
a64d49f49ca3 implement the reflexivity prover as a monotonicity prover that proves R >= op=; derive "reflexivity" rules for relators from mono rules and eq rules
kuncar
parents: 55083
diff changeset
   647
  Quotient_composition_ge_eq
a64d49f49ca3 implement the reflexivity prover as a monotonicity prover that proves R >= op=; derive "reflexivity" rules for relators from mono rules and eq rules
kuncar
parents: 55083
diff changeset
   648
  left_total_eq left_unique_eq left_total_composition left_unique_composition
a64d49f49ca3 implement the reflexivity prover as a monotonicity prover that proves R >= op=; derive "reflexivity" rules for relators from mono rules and eq rules
kuncar
parents: 55083
diff changeset
   649
  left_total_fun left_unique_fun
51374
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   650
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   651
(* setup for the function type *)
47777
f29e7dcd7c40 use a quot_map theorem attribute instead of the complicated map attribute
kuncar
parents: 47698
diff changeset
   652
declare fun_quotient[quot_map]
51374
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   653
declare fun_mono[relator_mono]
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   654
lemmas [relator_distr] = pos_fun_distr neg_fun_distr1 neg_fun_distr2
47308
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   655
48891
c0eafbd55de3 prefer ML_file over old uses;
wenzelm
parents: 47982
diff changeset
   656
ML_file "Tools/Lifting/lifting_term.ML"
47308
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   657
48891
c0eafbd55de3 prefer ML_file over old uses;
wenzelm
parents: 47982
diff changeset
   658
ML_file "Tools/Lifting/lifting_def.ML"
47308
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   659
48891
c0eafbd55de3 prefer ML_file over old uses;
wenzelm
parents: 47982
diff changeset
   660
ML_file "Tools/Lifting/lifting_setup.ML"
47308
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   661
55563
a64d49f49ca3 implement the reflexivity prover as a monotonicity prover that proves R >= op=; derive "reflexivity" rules for relators from mono rules and eq rules
kuncar
parents: 55083
diff changeset
   662
hide_const (open) invariant POS NEG
47308
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   663
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   664
end